We study the explicit formula (suggested by Gamayun, Iorgov and Lisovyy) for the Painlevé III(D 8) τ function in terms of Virasoro conformal blocks with a central charge of 1. The Painlevé equation has two types of bilinear forms, which we call Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of the direct sum of two Virasoro algebras in a certain superalgebra. These two types of bilinear forms correspond to the Neveu–Schwarz sector and the Ramond sector of this algebra. We also obtain the τ functions of the algebraic solutions of the Painlevé III(D 8) from the special representations of the Virasoro algebra of the highest weight (n + 1/4)2.
We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.
We introduce and study a category (Formula presented.) of modules of the Borel subalgebra (Formula presented.) of a quantum affine algebra (Formula presented.), where the commutative algebra of Drinfeld generators (Formula presented.), corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional (Formula presented.) modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in (Formula presented.). Among them, we find the Baxter (Formula presented.) operators and (Formula presented.) operators satisfying relations of the form (Formula presented.). We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the (Formula presented.) operators acting in an arbitrary finite-dimensional representation of (Formula presented.).
We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between N=2 supersymmetric gauge theories and two-dimensional conformal field theory.Talk. 1 1 http://salafrancesco.altervista.org/wugo2015/tanzini.pdf. presented by A.T. at the conference Interactions between Geometry and Physics - in honor of Ugo Bruzzo's 60th birthday 17-22 August 2015, Guarujá, São Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).
We discuss the correspondence between the Knizhnik–Zamolodchikov equations associated with GL(N) and the n-particle quantum Calogero model in the case when n is not necessarily equal to N. This can be viewed as a natural 'quantization' of the quantum-classical correspondence between quantum Gaudin and classical Calogero models.
We consider stationary stochastic processes (Formula presented.) such that (Formula presented.) lies in the closed linear span of (Formula presented.); following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be linearly rigid, that the spectral density vanishes at zero and belongs to the Zygmund class (Formula presented.). We next give a sufficient condition for stationary determinantal point processes on (Formula presented.) and on (Formula presented.) to be linearly rigid. Finally, we show that the determinantal point process on (Formula presented.) induced by a tensor square of Dyson sine kernels is not linearly rigid.
For a point p of the complex projective plane and a triple (g,d,l) of non-negative integers we define a Hurwitz--Severi number H(g,d,l) as the number of generic irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as singularities at the other points, such that the projection of the curve from p has a prescribed set of local and remote tangents and lines passing through nodes. In the cases d+l >= g+2 and d+2l >= g+2 > d+l we express the above Hurwitz--Severi numbers via appropriate ordinary Hurwitz numbers. The remaining case d+2l < g+2 is still widely open.
We propose the q-deformation of the Gamayun-Iorgov-Lisovyy formula for the Painlevé τ function. Namely, we propose the formula for the τ function for the q-difference Painlevé equation corresponding to the surface (and the symmetry) in the Sakai classification. In this formula, the τ function equals the series of q-Virasoro Whittaker conformal blocks (equivalently, the Nekrasov partition functions for pure SU(2) 5d theory)
We study integrable models with gl(2|1) symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker than the Bethe equations. This representation allows us to find the norms of on-shell Bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix.